371 



i 

 !«: I, 



pu/ft, 



s 



TABLE 2 - THEORETICALLY CALCITLATED DYNAMIC PRESSURE 

 RESPONSE AT VARIOUS (x/c> LOCATIONS 



REDUCED FREQUENCY. K 

 0.3 0.5 0.75 



-7.47 



30.18 



.90 



-10.53 

 2B.10 



-7.46 -10.51 



28.03 26.09 



.93 .86 



At j/c - 0.0046. (- 



',). 



33.52 



-11.25 



22.41 

 .67 



-9.53 

 21.87 



-7.91 



20.97 



.63 



-6.97 -6.26 



20.57 20.27 



.61 .61 



At x/c 



i 0.0073, (t^I - 30.25 



-11.11 



21.72 



.72 



-9.25 

 20.28 



-7.45 



19.44 



.65 



-6.32 -5.26 



19.06 18.77 



.63 .62 



dc 

 At x/c - 0.0117. (t^)_ - 26.59 



At x/c - 0.018, (j-D - 23.0 



-5.96 



20.19 



.60 



-4.61 

 18.68 



Ma's 



-6.66 -5.22 -3.56 -2.31 



16.77 16.44 16.19 16.10 



.63 .62 .61 .61 



dc 



relationship is not realized in the present test 

 results (See Figure 7) . This kind of discrepancy 

 in applying the above scaling law for cavitation 

 inception is a classic problem and has been exten- 

 sively discussed in the literature [for example see 

 Morgan and Peterson (1977) and Acosta and Parkin 

 (1975)]. 



One of the possible reasons for this discrepancy 

 is that a finite amount of time is required for 

 nuclei to grow. Thus, cavitation inception will 

 depend not only on the magnitude of the suction 

 pressure peak, but it will also depend on the shape 

 of the pressure distribution in the neighborhood 

 of the suction peak and the peak location. Since, 

 as shown previously, these two features of the 

 pressure distribution are essentially the same for 

 zero and nonzero reduced frequencies of interest 

 here, it will be assumed that the amount of time 

 required for nuclei to grow is approximately the 

 same for both a stationary and oscillating foil. 

 Consequently, it is assumed that cavitation incep- 

 tion occurs on the foil at nonzero reduced 

 frequencies when the magnitude of -Cpsn,in '°is' -"-^ 

 encountered during the foil oscillation, for given 

 values of a and Rn. 



An analytical method will now be developed to 

 predict leading edge cavitation inception on a 

 oscillating foil based on inception measurements 

 made on a stationary foil. Let ACp be given by 



ISJ/a, 



dc 



At x/c - 0.026. (j-^)^ • 19.65 



-7.37 -10.28 -10.08 -7.19 -4.01 -1.52 +2.15 



17.72 16.47 13.63 12.67 12.12 11.88 11.71 



5 .90 .84 .69 .65 .62 .61 .60 



dc 

 At x/c = 0.035, (x;J). - 16.79 



+5.40 



11.68 



.60 



Ac = C„ . 

 p ' Psmin 



(a. ) 

 is 



psmm 



(a ) 



(10) 



where Cpgj,,^^ (oIq' denotes the minimum value of the 

 static pressure coefficient at a = Uq and Cpgjoj^jj (ciis' 

 is the minimum static pressure coefficient at the 

 cavitation inception angle ci-j^g . According to our 

 assumption, unsteady cavitation occurs when the 

 difference in the static loading ACp between a^^, 

 and Oq is produced by the dynamic loading at some 

 instant of time t j^ . Thus, unsteady cavitation 

 occurs if 



I C (t.) I = ACp (11) 



pu 1 '^ 



l%ul/a, 



-7.18 -9.84 



11.68 10.84 



.92 .85 



At x/c - 0,058. (j^), - 12.78 



where |Cpu(tj^) | is the magnitude of the dynamic 

 pressure response at time t = t ^^ . If the value of 

 "is ~ "o ^^ small it follows from Eqs. (5) and (11) 

 that 



ai (dC /da) sin C^t. + 

 '^ p u 1 



-.[a - a ) (dC /dC) 

 IS p s 



(12) 



and nonzero reduced frequencies. This is an 



important conclusion which will be utilized later 



in the analytical prediction of cavitation inception. 



We will now proceed to develop a criterion to 

 define the unsteady leading edge cavitation inception. 

 Let c(j^3 denote the cavitation inception angle 

 measured in a stationary test for given values of 

 a and Rn. As an example, at a cavitation number of 

 o = 1.15 and Rn = 3 x 10 , cavitation inception 

 occurred experimentally at a-^s = 3.5°. The corre- 

 sponding pressure distribution calculated using 

 potential flow theory is given in Figure 7 with a 

 suction peak appearing at around 1.6 percent chord 



aft of the leading edge. Let C 



psmin 



("is 



denote 



the minimum value of the static pressure coefficient 

 Cps, at the foil angle a = a^^. It has been 

 generally assumed that cavitation inception occurs 

 when -Cpgjnijj ("is' ~ ^- Obviously, this simple 



where tj corresponds to that instant of time at 

 which Eq. (11) is satisfied. Small- amplitude 

 motion has been assumed. The static angular 

 pressure gradient is to be evaluated at the location 

 of the suction peak corresponding to the steady 

 condition a = a . The unsteady inception angle a^^ 

 for a given reduced frequency K is obtained from 

 Eqs. (1) and (12) . 



a + (a. 



IS 



a ) ^ 

 S 



a^ sin<i> \/l - (■ 



-) , (ai ^ 0) (13) 



ai S 



As a consequence of Eq. (12) , no singularity is 

 expected inside the square root. Due to the 



